The figure shows one layer of the three-dimensional structure of a Salt crystal.

Question

The figure shows one layer of the three-dimensional structure of a Salt crystal. The atoms extend much farther off in all directions, but only a six-by-six square is shown here. The larger circles are the chlorine ions, which have charges of -e, where e = 1.60 times 10^-19 C. The smaller circles are sodium ions, with charges of +e. The center-to-center distance between neighboring ions is about 0.3 nm. Real crystals are never perfect, and the crystal shown here has two defects: a missing atom one location, and an extra lithium atom, shown as a grey circle, inserted in one of the small gaps. If the lithium atom has a charge of +e, what is the direction and magnitude of the total force on it? Assume there are no other defects nearby in the crystal besides the two shown here.

Explanation / Answer

The force on the Li ion due to all Cl - ions is canceled since for every Cl attracting Li ion there is an equally distant one attracting it in opposite direction. However , in case of Na ions the repulsive force due to all Na ions cancel each other except the one corresponding to the gap where no Na ion is present. Consequently the pull due to corresponding opposite Na ion is not balanced and hence the repulsive force due the corresponding Na acts on Li ion.

Let us assume that Li ion is exactly in the Center of gap between Na and Cl ions.

Then the distance to unbalanced Na ion will be 0.3 nm/2 +0.6 nm ( = 0.75 nm)along horizontal direction and 0.3nm/2 + 0.6 nm (= 0.75 nm )

Hence the distance between Na and Li is = sqrt { 2 X (0.75)^2}

Now applying Coulomb's Law

F = K Q1 x Q2/ r^2 = 9.0 x 10^9 nt-m^2/coulombs^2 X 1.6 X 10^(-19) C X 1.6 X 10^(-19) C/ (2 X 0.75^2 X 10^(-18))

=, 20.48 X 10^(-21) newtons

This force will be in the direction of line joing the gap defect and the Li ion.

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